The longer an organism lives the more it can reproduce, the more copies of it's genes there will be. Any gene that increases (reproductive) lifespan is strongly selected for.
Not correct, though intuitive. Consider each offspring to be 50% of the genes of the parent. A parent with three children is competing with hir own genetics, and losing.
It seems like the rate of reproduction rather than total reproductive lifespan is what matters for selection. A longer lifespan at the expense of a lower rate of reproduction would not be selected for.